We study disordered capillary waves on the surface of a vertically oscillated fluid layer and try to establish their relation with weak wave turbulence. We measure the surface gradient in space and time and argue that gradient spectra are better suited for comparison to the predictions of weak wave turbulence theory than spectra of the surface elevation. Because the gradient is a vector quantity, we must distinguish longitudinal and transverse spectra. However, they prove to be related trivially through isotropy. From the measured wavenumber-frequency spectrum it appears that the dispersion relation is only satisfied approximately. In the wavenumber direction the spectral features are strongly broadened due to spatial disorder. This disagrees with weak wave turbulence theory where exact satisfaction of the dispersion relation is pivotal. We find approximate algebraic frequency and wavenumber spectra but with exponents that are different from those predicted by weak wave turbulence theory. However, other findings, such as the broadening of harmonics proportional to their frequency and the emergence of Gaussian statistics at selected wavenumber bands, point to weak wave turbulence.

We study dynamical behavior of local structures, such as sources and holes, in traveling-wave patterns in a very long ͑2 m͒ heated wire convection experiment. The sources undergo a transition from stable coherent behavior to erratic behavior when the driving parameter is decreased. This transition, as well as the scaling of the average source width in the erratic regime are both qualitatively and quantitatively in accord with earlier theoretical predictions. We also present results for the holes sent out by the erratic sources. DOI: 10.1103/PhysRevE.67.036305 PACS number͑s͒: 47.20.Bp, 07.05.Kf, 47.20.Dr, 47.54.ϩr Traveling-wave systems play an exceptional role within the field of pattern formation. If the transition to patterns is supercritical ͑forward͒, the dynamics close to threshold should be amenable to a description by the complex Ginzburg-Landau ͑CGL͒ amplitude equation ͓1͔. Theory and experiments are difficult to compare, however, for the following two reasons.͑i͒ Both the CGL model and experimentally observed traveling-wave patterns exhibit an astonishing variety of ordered, disordered, and chaotic dynamics, which can be difficult to characterize or compare.͑ii͒ The dynamics depends, in general, strongly on nonuniversal coefficients ͓2-4͔, but the values of these coefficients are difficult to determine in experiments ͓5-7͔.The study of local structures, such as sources, fronts, and holes, which play an important role in traveling-wave systems ͓1-4,7-12͔, provide a promising route to compare theory and experiment as they partially circumvent these difficulties: their nontrivial behavior often depends only on a subset of the coefficients ͓11͔ and is, in addition, relatively easy to characterize experimentally ͓7-9,12͔.In this paper, we present a successful example of this approach in a heated wire convection experiment ͑Fig. 1͒. This system forms left and right traveling waves that suppress each other; typical states consist of patches of left and right traveling waves separated by sources ͑which send out waves͒ and sinks ͑which have two incoming waves͒ ͓13͔. An earlier theoretical work ͓14͔, which was based on the amplitude equations ͑1͒ and ͑2͒ below, predicts that, essentially due to the transition from an absolute to a convective instability ͓15͔, sources tend to display a diverging width when the driving parameter is lowered beyond a critical value ͓Eq. ͑3͔͒. More recently, it was predicted that just before these stationary sources would diverge, they become unstable and give way to fluctuating sources of finite average width which display highly nontrivial dynamics ͓4͔.We indeed observe this nontrivial change in source behavior when the driving ͑heating of wire͒ is decreased; not only the measured transition value, but also the qualitative behavior of sources is in accord with the predictions ͓14,4͔. All properties necessary to compare theory and experiment are measured in a set of independent experiments. The fluctuating sources send out holes, and we show that these display behavior very simil...

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